Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space

TL;DR

This paper shows that most states in Hilbert space are physically inaccessible within polynomial time using local Hamiltonians, implying limitations on quantum simulation and the equivalence of dynamic Hamiltonian models to standard quantum circuits.

Contribution

It introduces a time-dependent Suzuki-Trotter expansion and counting argument to demonstrate the exponential smallness of physically realizable states in Hilbert space.

Findings

Most states are not physically accessible within polynomial time.

Rapidly changing Hamiltonians do not surpass standard quantum circuit models.

The volume of states reachable by local Hamiltonians is exponentially small.

Abstract

We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space. This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time. We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a counting argument. This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model.